Is there a theoretically interesting notion that is a natural extension of the concept of rigidity to general terms?

Such a notion ought to satisfy two Kripkean conditions.

First, it must apply to typical general terms for natural kinds1, stuffs, and phenomena, and fail to apply to most other general terms.

Second, true 'identification sentences' (such as 'Cats are animals') containing general terms that the notion applies to must be necessary.

I explore a natural extension of the notion of rigidity to general terms, the notion of an essentialist predicate. I argue that, under natural assumptions, this notion satisfies the two Kripkean conditions.